16-Mar-2015/18-Mar-2015: Finding the coefficient of friction

PURPOSE

The purpose of this experiment was to use various experimental and mathematical methods to find the coefficient of friction between two surfaces.

PROCEDURES

Part 1: Static Friction

Figure 1: Initial set-up of our experiment

We began the first part of this experiment with the set-up shown in Figure 1 (except we started with one block, not two). We added water to the cup until the block began to accelerate. We then measured the mass of the water and the cup that caused the equilibrium to be broken. We also measured the mass of the block. We repeated this step three times, adding another block each time. The collected data is shown in Figure 2.

Figure 2: Data set

Also displayed in Figure 2 is the normal force acting on the block. We found this by setting up Newton's second law of motion in the y-direction. Since the block was not accelerating vertically, we set the normal force of the block equal to its weight. In other words, we found column three of Figure 2 by multiplying the corresponding values of column one with the acceleration due to gravity: 9.81 m/s² (we used 9.81 m/s² instead of 9.8 m/s² in order to be more accurate with our calculations). After we found the normal force, we determined the maximum static friction force. We accomplished this task by first assuming that the tension in the string that was pulling on the block was equal to the weight of the water and cup system. Then, we assumed that this tension was also equal to the maximum static friction force because it is the amount of force that was required to bring the block out of equilibrium. From these calculated values, we constructed a graph with the friction force on the y-axis and the normal force on the x-axis. The slope of this graph was intended to give us the coefficient of static friction (µ_s), which was 0.2068 in this case. The graph is illustrated in Figure 3 (click to enlarge).

Figure 3: Static friction vs normal force graph

Part 2: Kinetic Friction

After solving for the coefficient of static friction between the block and the track, we attempted to solve for the coefficient of kinetic friction (µ_k) of the same system. To accomplish this task, we used a set-up that was very similar to the one shown in Figure 1. The only modification we made was that we replaced the pulley and cup with a force sensor. We pulled on this force at a constant rate to implement a constant force on the block. Then, we repeated the process three more times, adding another block on top of the first after each trial. The forces that we applied on the blocks are shown below in Figure 4 (click to enlarge).

Figure 4: Graphs of the forces applied on the blocks during each trial

We took the averages of the graphs in Figure 4 to find the kinetic friction force. We once again plotted these forces against the normal forces for each block to find the coefficient of friction from the slope of the graph. We found this value to be 0.2168, which was actually bigger than the coefficient of static friction found earlier. The reason why this may have been so will be discussed later in the conclusion. This graph is displayed in Figure 5 (click to enlarge).

Figure 5: Kinetic friction vs normal force graph

Part 3: Static Friction from a Sloped Surface

For the third part of the experiment, we sought to find the coefficient of static friction by utilizing a different method. We once again put the block on the track and increased the track's slope until the block began to move. We measured the angle of this slope, which was 15°, and solved for µ_s by plugging in this angle into Equation 1:

f_s = m*g*sinθ

=> µ_s*N = m*g*sinθ

=> µ_s*m*g*cosθ = m*g*sinθ

=> µ_s = sinθ/cosθ = tanθ (Equation 1)

where θ is the angle between the track and the horizontal surface.

The resulting value was 0.268. We could not compare this value to the value found in Part 1 of the experiment because we used different blocks.

Part 4: Kinetic Friction from Sliding a Block Down an Incline

Figure 6: Set-up for part 4 of the lab

Next, we implemented the set-up shown in Figure 6 with the track at an angle of 23° with the horizontal. We then placed the block on the track and allowed it to accelerate. We captured the block's velocity as it went down the track with the motion sensor. Then, we applied a Linear Fit on the velocity graph in Logger Pro to find the block's acceleration, which was 1.328 m/s². This is shown in Figure 7 (click to enlarge).

Figure 7: Velocity vs time graph (above)

We used this acceleration to solve for µ_k by plugging the acceleration into Equation 2:

m*g*sinθ - f_k = m*a, where f_k = µ_k*m*g*cosθ (Equation 2).

m*g*sinθ - µ_k*m*g*cosθ = m*a
µ_k*m*g*cosθ = m*g*sinθ - m*a
µ_k = (g*sinθ - a)/(g*cosθ)

When we plugged in the corresponding values, we found µ_k to be 0.277. Once again, this was bigger than the µ_s we found in the Part 3 of the experiment.

Part 5: Predicting the Acceleration of a Two-Mass System

Finally, we utilized the µ_k found in the previous section to predict the acceleration of a block when we applied a pull force on it from a hanging mass and pulley system. In order to do so, we used an equation similar to Equation 2:

m_2*g - f_k = m_1*a (Equation 3), where f_k = µ_k*m_1*g and m_1 and m_2 are the masses of the block and the hanging mass, respectively.

Plugging in the respective values, we found the acceleration to be 0.506 m/s². Then, we found the acceleration from the slope of the velocity versus time graph in Logger Pro (Figure 8). We found this value to be 0.2671 m/s².

Figure 8: Velocity vs time graph 2

CONCLUSION

When we solved for the coefficient of kinetic friction, we noticed that it was larger than the coefficient of static friction, both times. This result is the opposite of the outcome that we expected. One possible reason for this is that bottom of the block and the track are not completely uniform surfaces, which may have given us unreliable data. Another factor that could have contributed to this inaccurate result is the "constant" force that we applied on the block in the second step; this force was most likely variable and not constant. Furthermore, we noticed that the accelerations that we found in Part 5 of the experiment were different by a considerable amount. When we solved for the acceleration using mathematical methods, we found it to be 0.506 m/s². On the other hand, experimentally, we found the acceleration to be 0.2671 m/s². The percent error between the theoretical and experimental values was 47.21 percent. We believe that the disparity in these results is due to similar reasons discussed. Any irregularities on the bottom of the block or the track could have slowed down the block's acceleration by a significant amount as these were not considered in our calculations. In addition, we assumed that the pulley we used was frictionless, even though it was not. Although the pulley most likely did not have as great as an affect as the irregular surfaces since it was very close to being frictionless, it still could have been a factor.